Search for the $\Lambda_c\Sigma_c$ and $\bar{\Lambda}_c\Sigma_c$ dibaryon structures via the QCD sum rules

Using QCD sum rules, this paper constructs hexaquark currents to investigate $\Lambda_c\Sigma_c$ and $\bar{\Lambda}_c\Sigma_c$ dibaryon structures, identifying three potential molecular bound states ($\Lambda_c\Sigma_c$ with $J^P=1^+$ and $\bar{\Lambda}_c\Sigma_c$ with $J^P=0^-$ and $1^-$) while classifying the remaining five configurations as resonance states.

The Gist

Imagine the universe as a giant, cosmic LEGO set. For decades, physicists have been building structures with these tiny blocks called quarks.

Usually, the rules are simple:

• Mesons are like small pairs (two blocks stuck together).
• Baryons (like protons and neutrons) are like small triangles (three blocks stuck together).

But recently, scientists have started finding weird, exotic structures that don't fit these simple shapes. They are looking for Hexaquarks—structures made of six quarks stuck together. Think of these as complex, six-block LEGO towers.

This paper is a theoretical investigation into two specific types of these six-block towers: ones made from a $\Lambda_c$ and a $\Sigma_c$ (and their anti-matter twins). The authors are asking: Do these six blocks stick together tightly to form a stable molecule, or do they just bounce off each other like magnets with the same pole?

Here is the breakdown of their journey, explained simply:

1. The Detective's Toolkit: QCD Sum Rules

The authors can't build these towers in a lab yet because they are too heavy and unstable. Instead, they use a mathematical detective tool called QCD Sum Rules.

Think of this tool like a high-tech metal detector.

• You can't see the treasure (the particle) underground.
• But you can scan the ground (the vacuum of space) with your detector.
• If the detector beeps in a specific pattern, you know there's a treasure chest buried there, and you can even guess its weight and size.

In this case, the "ground" is the quantum vacuum, and the "beeps" are mathematical signals called currents. The authors built eight different types of metal detectors (mathematical formulas) to scan for these six-quark structures.

2. The Search: Two Types of Towers

They were looking for two main combinations:

1. $\Lambda_c \Sigma_c$: A mix of two specific types of heavy quark baryons.
2. $\bar{\Lambda}_c \Sigma_c$: A mix involving an anti-matter baryon.

They checked these towers in four different "orientations" (called Spin and Parity, or $J^P$). Imagine trying to stack the LEGO blocks in four different ways:

• Flat and calm ($0^+$)
• Flat and spinning ($0^-$)
• Standing up ($1^+$)
• Standing up and spinning ($1^-$)

3. The Results: Who Sticks and Who Bounces?

After running their complex calculations, the "metal detector" gave them a list of results. They found that out of the eight possibilities they checked, only three seem to be stable enough to be called "molecules" (where the blocks are glued together). The other five are just "resonances" (where the blocks bump into each other and fly apart).

Here is the verdict on the specific towers:

• The "Glued" Towers (Molecular States):

  • The $\Lambda_c \Sigma_c$ tower with a specific spin ($1^+$) seems to stick together. It's a bit lighter than the two separate pieces, meaning it's a stable molecule.
  • The $\bar{\Lambda}_c \Sigma_c$ towers with spins $0^-$ and $1^-$ also seem to stick together.
  • Analogy: These are like two magnets that have been glued together. They form a single, new object.

• The "Bouncing" Towers (Resonance States):

  • The other five combinations (like the $\Lambda_c \Sigma_c$ with spin $0^-$ or $0^+$) are too heavy. They are heavier than the two separate pieces combined.
  • Analogy: These are like two magnets that are repelling each other. They might touch for a split second, but they immediately fly apart. They aren't stable molecules; they are just temporary collisions.

4. Why Does This Matter?

You might ask, "Why build a math model of a particle we haven't seen yet?"

• The Puzzle of the Universe: We know about protons and neutrons, but the universe is full of "exotic" matter. Understanding how six quarks stick together helps us understand the strong force—the glue that holds the universe together.
• The Experimental Challenge: The paper mentions that a real-world experiment (BESIII) recently looked for these particles but didn't see them yet. This isn't a failure; it's a clue! It tells experimentalists, "Hey, if you look for these specific heavy particles, you might find them. But if you look for the lighter ones, you probably won't."
• The Future: If these "molecular" towers exist, they could be the missing pieces in our understanding of how matter is formed, perhaps even helping us understand the inside of neutron stars.

Summary

In short, these scientists used advanced math to simulate building six-quark LEGO towers. They found that three specific designs are stable enough to exist as new particles (molecules), while the other five designs fall apart immediately. This gives experimental physicists a "Wanted Poster" with the exact weight and shape of the particles they should try to find in the real world.

Technical Summary

1. Problem Statement

The paper addresses the theoretical search for hexaquark dibaryon states composed of charmed baryons, specifically the $\Lambda_c\Sigma_c$ and $\bar{\Lambda}_c\Sigma_c$ systems. While the deuteron and the $d^*(2380)$ are the only experimentally confirmed genuine dibaryons, the existence of other heavy-flavor dibaryons (molecular states or compact hexaquarks) remains an open question in Quantum Chromodynamics (QCD).

Recent experimental efforts by the BESIII collaboration to search for these states near threshold energies ($4.715 - 4.951$ GeV) have yielded no signals, necessitating robust theoretical predictions to guide future searches. The authors aim to determine the mass spectrum, quantum numbers ($J^P$), and binding nature (bound molecular state vs. unbound resonance) of these hidden-charm dibaryons.

2. Methodology

The authors employ the QCD Sum Rules (QCDSR) method, a non-perturbative approach that relates hadronic properties to QCD vacuum condensates via the Operator Product Expansion (OPE).

• Interpolating Currents:
  The authors constructed eight pairs of local six-quark interpolating currents to represent the $\Lambda_c\Sigma_c$ and $\bar{\Lambda}_c\Sigma_c$ systems. These currents cover four possible spin-parity configurations: $J^P = 0^-, 0^+, 1^+, 1^-$.

  • The currents are built from the known currents of the $\Lambda_c$ and $\Sigma_c$ baryons ($J_\Lambda$ and $J_\Sigma$).
  • The authors demonstrated that within each pair (e.g., $J_1$ and $J'_1$), the currents are equivalent, allowing them to select one representative current per pair for calculation.

• Correlation Functions:
  Two-point correlation functions were calculated on two sides:

  1. Hadronic Side: Saturated by a complete set of intermediate hadronic states (ground states and continuum), introducing parameters for mass ($M$) and pole residue ($\lambda$).
  2. QCD Side: Calculated using the OPE up to dimension 12. This involves contracting quark fields using full light ($u, d$) and heavy ($c$) quark propagators.

• Sum Rule Derivation:

  • The Borel transform was applied to both sides to suppress higher-state contributions and enhance the ground state signal.
  • Continuum Threshold ($s_0$): Determined empirically as $\sqrt{s_0} = M_Z + (0.5 \sim 0.7)$ GeV.
  • Energy Scale ($\mu$): Determined using the formula $\mu = \sqrt{M_X^2 - 4M_c^2}$, where $M_c$ is the effective charm quark mass ($1.84$ GeV).
  • Validity Criteria: The results were extracted from "Borel windows" satisfying two conditions:
    1. Pole Dominance: The ground state contribution must be significant (typically $>40\%$).
    2. OPE Convergence: Higher-dimensional condensate contributions must decrease rapidly.

3. Key Contributions

• Systematic Construction: The paper provides a comprehensive construction of eight pairs of hexaquark currents specifically for the $\Lambda_c\Sigma_c$ and $\bar{\Lambda}_c\Sigma_c$ systems, covering all relevant $J^P$ combinations without orbital angular momentum.
• Equivalence Proof: The authors rigorously proved the equivalence of the constructed current pairs, simplifying the analysis while ensuring no physical states are missed.
• High-Dimensional OPE: The calculation includes vacuum condensates up to dimension 12, ensuring high precision in the theoretical prediction.
• Differentiation of States: The study explicitly distinguishes between bound molecular states (mass below the two-baryon threshold) and resonance states (mass above the threshold).

4. Key Results

The authors extracted the masses and pole residues for the ground states. The threshold for the $\Lambda_c + \Sigma_c$ system is approximately $4.74$ GeV ($2.285 + 2.455$ GeV).

A. $\Lambda_c\Sigma_c$ System:

• $J^P = 1^+$ (Current $J_3$): Mass $M = 4.73^{+0.08}_{-0.08}$ GeV.
  • Conclusion: The central value is $10$ MeV below the threshold. Given the uncertainty, this is identified as a possible molecular bound state.
• $J^P = 0^-$ (Current $J_1$): Mass $M = 5.60$ GeV.
  • Conclusion: Well above threshold; identified as a resonance state.
• $J^P = 0^+$ (Current $J_2$): Mass $M = 4.86$ GeV.
  • Conclusion: Above threshold; identified as a resonance state.
• $J^P = 1^-$ (Current $J_4$): Mass $M = 4.88$ GeV.
  • Conclusion: Above threshold; identified as a resonance state.

B. $\bar{\Lambda}_c\Sigma_c$ System:

• $J^P = 0^-$ (Current $J_6$): Mass $M = 4.69^{+0.09}_{-0.09}$ GeV.
  • Conclusion: Below threshold; identified as a possible molecular bound state.
• $J^P = 1^-$ (Current $J_7$): Mass $M = 4.72^{+0.09}_{-0.08}$ GeV.
  • Conclusion: Near or slightly below threshold; identified as a possible molecular bound state.
• $J^P = 0^+$ (Current $J_5$): Mass $M = 5.10$ GeV.
  • Conclusion: Above threshold; identified as a resonance state.
• $J^P = 1^+$ (Current $J_8$): Mass $M = 5.18$ GeV.
  • Conclusion: Above threshold; identified as a resonance state.

Pole Residues:
All calculated pole residues ($\lambda$) are of the order of magnitude $10^{-3}$ GeV$^8$, indicating a reasonable coupling strength between the currents and the physical states.

5. Significance

• Theoretical Guidance: The paper provides specific mass predictions and quantum numbers for potential dibaryon states, offering concrete targets for experimentalists (e.g., BESIII, LHCb, Belle II) to search for signals in specific energy windows.
• Nature of Exotic States: It reinforces the hypothesis that while many heavy-flavor hexaquark configurations are likely unbound resonances, specific channels (specifically $\Lambda_c\Sigma_c$ with $1^+$ and $\bar{\Lambda}_c\Sigma_c$ with $0^-, 1^-$) are strong candidates for hadronic molecules.
• Contextualizing Null Results: The findings help interpret the lack of signals in previous BESIII searches; if the bound states exist, they may be narrow or located in specific channels not yet fully explored, while the resonance states might be too broad to be easily distinguished from the continuum.
• Methodological Benchmark: The work serves as a benchmark for applying QCD sum rules to heavy-flavor hexaquark systems, demonstrating the importance of high-dimensional condensates and careful energy scale determination.